For n = 1, there is one 1cell pattern because there is only one unit square to paint.
For n = 2, there are two 2cell patterns:
+++ +++ ++
 1  2   1  2   1 
+++ +++ and +++
 3  4   4 
+++ ++
For n = 3, there is one 3cell pattern:
++++
 1  2  3 
++++
 4  5  6  it is ++++
++++  1  2  3 
 7  8  9  ++++
++++
For n = 4, there are sixty 4cell patterns:
+++++
 1  2  3  4 
+++++
 5  6  7  8  one is +++++
+++++  1  2  3  4 
 9 10 11 12  +++++
+++++
13 14 15 16 
+++++
+++++ ++
 1  2  3  4  is equivalent to  1 
+++++ ++
 5 
++
 9 
++
13 
++
and therefore is counted as one pattern.
Another 4cell pattern for a 4 X 4
+++++
 x  x  y  y 
+++++
 z  y  x  a  is +++
+++++  x  x 
 y  z  a  x  ++++
+++++  x 
 a  a  z  z  +++
+++++  x 
++
+++
 x  x 
++++ is equivalent to
 x 
+++
 x 
++
+++ ++ ++
 y  y   z   a 
++++ +++ +++
 y   z   a 
+++ ++++ ++++
 y   z  z   a  a 
++ +++ +++
because the shapes can be created through reflection, rotation, or translation.
Therefore, they are counted as one pattern.
For n = 5, there is one 5cell pattern.
